Problem: Determine the value of the following complex number power. Your answer will be plotted in orange. $ ({\cos(\frac{5}{12}\pi) + i \sin(\frac{5}{12}\pi)}) ^ {9} $
Answer: Let's express our complex number in Euler form first. $ {\cos(\frac{5}{12}\pi) + i \sin(\frac{5}{12}\pi)} = { e^{5\pi i / 12}} $ Since $(a ^ b) ^ c = a ^ {b \cdot c}$ $ ({ e^{5\pi i / 12}}) ^ {9} = e ^ {9 \cdot (5\pi i / 12)} $ The angle of the result is $9 \cdot \frac{5}{12}\pi$ , which is $\frac{15}{4}\pi$ Our result is $ e^{7\pi i / 4}$. Converting this back from Euler form, we get $\cos(\frac{7}{4}\pi) + i \sin(\frac{7}{4}\pi)$.